Theoretical Statistics and Mathematics Unit, ISI Delhi

February 18, 2014 (Tuesday) ,
3:30 PM at Webinar

Speaker:
Yuri Bilu,
University of Bordeaux, France

Title:
Drawing curves on checkered paper.

Abstract of Talk

Imagine a sheet of checkered paper, the one used in arithmetic classes in primary schools. Let us try to draw a curve on this sheet which would intersect as many ``crossings'' as possible. (In other words: how many lattice points a compact curve can meet?) Of course, one can draw a rather ``curvy'' curve which would pass through every crossing. But the problem becomes interesting if the curve is assumed ``not too curvy''. For instance, in 1927 the Czech mathematician Jarnik proved that a *strictly convex* curve can pass at most $O(N^{2/3})$ crossings on an $N \times N$ checkered sheet. I will prove the theorem of Jarnik using an argument suggested by the German mathematician Dörge (1926), who worked independently of Jarnik on related problems. I will then show how a slight modification of D\"orge's argument leads to a wonderful theorem of Bombieri and Pila (1989) on counting lattice points on analytic curves.
If time permits, I will say a few words how in the last years these ideas were successfully used by Zannier, Pila and others to attack some prominent conjectures.